\(\int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {3+2 \cos (c+d x)}} \, dx\) [648]

Optimal result

Integrand size = 25, antiderivative size = 58 \[ \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {3+2 \cos (c+d x)}} \, dx=\frac {2 \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {3+2 \cos (c+d x)}}{\sqrt {5} \sqrt {\cos (c+d x)}}\right ),-5\right ) \sqrt {-\tan ^2(c+d x)}}{d} \]

[Out]

2*cot(d*x+c)*EllipticF(1/5*(3+2*cos(d*x+c))^(1/2)*5^(1/2)/cos(d*x+c)^(1/2),I*5^(1/2))*(-tan(d*x+c)^2)^(1/2)/d

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {2894} \[ \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {3+2 \cos (c+d x)}} \, dx=\frac {2 \sqrt {-\tan ^2(c+d x)} \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2 \cos (c+d x)+3}}{\sqrt {5} \sqrt {\cos (c+d x)}}\right ),-5\right )}{d} \]

[In]

Int[1/(Sqrt[Cos[c + d*x]]*Sqrt[3 + 2*Cos[c + d*x]]),x]

[Out]

(2*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[3 + 2*Cos[c + d*x]]/(Sqrt[5]*Sqrt[Cos[c + d*x]])], -5]*Sqrt[-Tan[c + d*x
]^2])/d

Rule 2894

Int[1/(Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[-2*S
qrt[a^2]*(Sqrt[-Cot[e + f*x]^2]/(a*f*Sqrt[a^2 - b^2]*Cot[e + f*x]))*Rt[(a + b)/d, 2]*EllipticF[ArcSin[Sqrt[a +
b*Sin[e + f*x]]/Sqrt[d*Sin[e + f*x]]/Rt[(a + b)/d, 2]], -(a + b)/(a - b)], x] /; FreeQ[{a, b, d, e, f}, x] &&
GtQ[a^2 - b^2, 0] && PosQ[(a + b)/d] && GtQ[a^2, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {2 \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {3+2 \cos (c+d x)}}{\sqrt {5} \sqrt {\cos (c+d x)}}\right ),-5\right ) \sqrt {-\tan ^2(c+d x)}}{d} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(140\) vs. \(2(58)=116\).

Time = 1.16 (sec) , antiderivative size = 140, normalized size of antiderivative = 2.41 \[ \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {3+2 \cos (c+d x)}} \, dx=\frac {4 \sqrt {\cos (c+d x)} \sqrt {3+2 \cos (c+d x)} \sqrt {-\cot ^2\left (\frac {1}{2} (c+d x)\right )} \csc (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {(3+2 \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}}{\sqrt {6}}\right ),6\right )}{d \sqrt {-\cos (c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {(3+2 \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}} \]

[In]

Integrate[1/(Sqrt[Cos[c + d*x]]*Sqrt[3 + 2*Cos[c + d*x]]),x]

[Out]

(4*Sqrt[Cos[c + d*x]]*Sqrt[3 + 2*Cos[c + d*x]]*Sqrt[-Cot[(c + d*x)/2]^2]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[(3
+ 2*Cos[c + d*x])*Csc[(c + d*x)/2]^2]/Sqrt[6]], 6])/(d*Sqrt[-(Cos[c + d*x]*Csc[(c + d*x)/2]^2)]*Sqrt[(3 + 2*C
os[c + d*x])*Csc[(c + d*x)/2]^2])

Maple [A] (verified)

Time = 7.25 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.79

method	result	size
default	\(-\frac {\left (1+\cos \left (d x +c \right )\right ) \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {10}\, \sqrt {\frac {3+2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, F\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), \frac {i \sqrt {5}}{5}\right )}{5 d \sqrt {3+2 \cos \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right )}}\)	\(104\)

[In]

int(1/cos(d*x+c)^(1/2)/(3+2*cos(d*x+c))^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/5/d*(1+cos(d*x+c))*2^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)/(3+2*cos(d*x+c))^(1/2)*10^(1/2)*((3+2*cos(d*x+
c))/(1+cos(d*x+c)))^(1/2)*EllipticF(cot(d*x+c)-csc(d*x+c),1/5*I*5^(1/2))/cos(d*x+c)^(1/2)

Fricas [F]

\[ \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {3+2 \cos (c+d x)}} \, dx=\int { \frac {1}{\sqrt {2 \, \cos \left (d x + c\right ) + 3} \sqrt {\cos \left (d x + c\right )}} \,d x } \]

[In]

integrate(1/cos(d*x+c)^(1/2)/(3+2*cos(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(2*cos(d*x + c) + 3)*sqrt(cos(d*x + c))/(2*cos(d*x + c)^2 + 3*cos(d*x + c)), x)

Sympy [F]

\[ \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {3+2 \cos (c+d x)}} \, dx=\int \frac {1}{\sqrt {2 \cos {\left (c + d x \right )} + 3} \sqrt {\cos {\left (c + d x \right )}}}\, dx \]

[In]

integrate(1/cos(d*x+c)**(1/2)/(3+2*cos(d*x+c))**(1/2),x)

[Out]

Integral(1/(sqrt(2*cos(c + d*x) + 3)*sqrt(cos(c + d*x))), x)

Maxima [F]

\[ \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {3+2 \cos (c+d x)}} \, dx=\int { \frac {1}{\sqrt {2 \, \cos \left (d x + c\right ) + 3} \sqrt {\cos \left (d x + c\right )}} \,d x } \]

[In]

integrate(1/cos(d*x+c)^(1/2)/(3+2*cos(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(2*cos(d*x + c) + 3)*sqrt(cos(d*x + c))), x)

Giac [F]

\[ \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {3+2 \cos (c+d x)}} \, dx=\int { \frac {1}{\sqrt {2 \, \cos \left (d x + c\right ) + 3} \sqrt {\cos \left (d x + c\right )}} \,d x } \]

[In]

integrate(1/cos(d*x+c)^(1/2)/(3+2*cos(d*x+c))^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(2*cos(d*x + c) + 3)*sqrt(cos(d*x + c))), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {3+2 \cos (c+d x)}} \, dx=\int \frac {1}{\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {2\,\cos \left (c+d\,x\right )+3}} \,d x \]

[In]

int(1/(cos(c + d*x)^(1/2)*(2*cos(c + d*x) + 3)^(1/2)),x)

[Out]

int(1/(cos(c + d*x)^(1/2)*(2*cos(c + d*x) + 3)^(1/2)), x)